1911 Encyclopædia Britannica/Napier, John

by James Whitbread Lee Glaisher

NAPIER, JOHN (1550–1617), Scottish mathematician and inventor of logarithms, was born at Merchiston near Edinburgh in 1550, and was the eighth Napier of Merchiston. The first Napier of Merchiston, “Alexander Napare,” acquired the Merchiston estate before the year 1438, from James I. of Scotland. He was provost of Edinburgh in 1437, and was otherwise distinguished. His eldest son Alexander, who succeeded him in 1454, was provost of Edinburgh in 1455, 1457 and 1469; he was knighted and held various important court offices under successive monarchs; at the time of his death in 1473 he was master of the household to James III. His son, John Napier of Rusky, the third of Merchiston, belonged to the royal household in the lifetime of his father. He also was provost of Edinburgh at various times, and it is a remarkable instance of the esteem in which the lairds of Merchiston were held that three of them in immediate lineal succession repeatedly filled so important an office during perhaps the most memorable period in the history of the city. He married a great-granddaughter of Duncan, 8th earl of Levenax (or Lennox), and besides this relationship by marriage the Napiers claimed a lineal male cadency from the ancient family of Levenax. His eldest son, Archibald Napier of Edinbellie, the fourth of Merchiston, belonged to the household of James IV. He fought at Flodden and escaped with his life, but his eldest son Alexander, (fifth of Merchiston) was killed. Alexander’s eldest son (Alexander, sixth of Merchiston) was born in 1513, and fell at the battle of Pinkie in 1547. His eldest son was Archibald, seventh of Merchiston, and the father of John Napier, the subject of this article.

In 1549 Archibald Napier, at the early age of about fifteen, married Janet, daughter of Francis Bothwell, and in the following year John Napier was born. In the criminal court of Scotland, the earl of Argyll, hereditary justice-general of the kingdom, sometimes presided in person, but more frequently he delegated his functions; and it appears that in 1561 Archibald Napier was appointed one of the justice-deputes. In the register of the court, extending over 1563 and 1564, the justice-deputes named are “Archibald Naper of Merchistoune, Alexander Bannatyne, burgess of Edinburgh, James Stirling of Keir and Mr Thomas Craig.” About 1565 he was knighted at the same time as James Stirling, his colleague, whose daughter John Napier subsequently married. In 1582 Sir Archibald was appointed master of the mint in Scotland, with the sole charge of superintending the mines and minerals within the realm, and this office he held till his death in 1608. His first wife died in 1563, and in 1572 he married a cousin, Elizabeth Mowbray, by whom he had three sons, the eldest of whom was named Alexander.^[1]

As already stated, John Napier was born in 1550, the year in which the Reformation in Scotland may be said to have commenced. In 1563, the year in which his mother died, he matriculated at St Salvator’s College, St Andrews. He early became a Protestant champion, and the one extant anecdote of his youth occurs in his address “to the Godly and Christian reader” prefixed to his Plaine Discovery. He writes:—

“In my tender yeares, and barneage in Sanct-Androis at the Schooles, having, on the one parte, contracted a loving familiaritie with a certaine Gentleman, &c. a Papist; And on the other part, being attentive to the sermons of that worthie man of God, Maister Christopher Goodman, teaching upon the Apocalyps, I was so mooved in admiration, against the blindnes of Papists, that could not most evidently see their seven hilled citie Rome, painted out there so lively by Saint John, as the mother of all spiritual whoredome, that not onely bursted I out in continual reasoning against my said familiar, but also from thenceforth, I determined with my selfe (by the assistance of Gods spirit) to employ my studie and diligence to search out the remanent mysteries of that holy Book: as to this houre (praised be the Lorde) I have bin doing at al such times as conveniently I might have occasion.”

The names of nearly all Napier’s classfellows can be traced as becoming determinantes in 1566 and masters of arts in 1568; but his own name does not appear in the lists. The necessary inference is that his stay at the university was short, and that only the groundwork of his education was laid there. Although there is no direct evidence of the fact, there can be no doubt that he left St Andrews to complete his education abroad, and that he probably studied at the university of Paris, and visited Italy and Germany. He did not, however, as has been supposed, spend the best years of his manhood abroad, for he was certainly at home in 1571, when the preliminaries of his marriage were arranged at Merchiston; and in 1572 he married Elizabeth, daughter of Sir James Stirling of Keir. About the end of the year 1579 his wife died, leaving him one son, Archibald (who in 1627 was raised to the peerage by the title of Lord Napier), and one daughter, Jane. A few years afterwards he married again, his second wife being Agnes, daughter of Sir James Chisholm of Cromlix, who survived him. By her he had five sons and five daughters.

In 1588 he was chosen by the presbytery of Edinburgh one of its commissioners to the General Assembly.

On the 17th of October 1593 a convention of delegates was held at Edinburgh at which a committee was appointed to follow the king and lay before him in a personal interview certain instructions relating to the punishment of the rebellious Popish earls and the safety of the church. This committee consisted of six members, two barons, two ministers and two burgesses—the two barons selected being John Napier of Merchiston and James Maxwell of Calderwood. The delegates found the king at Jedburgh, and the mission, which was a dangerous one, was successfully accomplished. Shortly afterwards another convention was held at Edinburgh, and it was resolved that the delegates sent to Jedburgh should again meet the king at Linlithgow and repeat their former instructions. This was done accordingly, the number of members of the committee being, however, doubled. These interviews took place in October 1593, and on the 29th of the following January Napier wrote to the king the letter which forms the dedication of the Plaine Discovery.

The full title of this first work of Napier’s is given below.^[2] It was written in English instead of Latin in order that “hereby the simple of this Iland may be instructed”; and the author apologizes for the language and his own mode of expression in the following sentences:—

“Whatsoever therfore through hast, is here rudely and in base language set downe, I doubt not to be pardoned thereof by all good men, who, considering the necessitie of this time, will esteem it more meete to make hast to prevent the rising againe of Antichristian darknes within this Iland, then to prolong the time in painting of language”; and “I graunt indeede, and am sure, that in the style of wordes and utterance of language, we shall greatlie differ, for therein I do judge my selfe inferiour to all men: so that scarcely in these high matters could I with long deliberation finde wordes to expresse my minde.”^[3]

Napier’s Plaine Discovery is a serious and laborious work, to which he had devoted years of care and thought. In one sense it may be said to stand to theological literature in Scotland in something of the same position as that occupied by the Canon Mirificus with respect to the scientific literature, for it is the first published original work relating to theological interpretation, and is quite without a predecessor in its own field. Napier lived in the very midst of fiercely contending religious factions; there was but little theological teaching of any kind, and the work related to what were then the leading political and religious questions of the day.

After the publication of the Plaine Discovery, Napier seems to have occupied himself with the invention of secret instruments of war, for in the Bacon collection at Lambeth Palace there is a document, dated the 7th of. June 1596 and signed by Napier, giving a list of his inventions for the defence of the country against the anticipated invasion by Philip of Spain. The document is entitled “Secrett Inventionis, proffitabill and necessary in theis dayes for defence of this Iland, and withstanding of strangers, enemies of God’s truth and religion,”^[4] and the inventions consist of (1) a mirror for burning the enemies’ ships at any distance, (2) a piece of artillery destroying everything round an arc of a circle, and (3) a round metal chariot, so constructed that its occupants could move it rapidly and easily, while firing out through small holes in it. It has been asserted (by Sir Thomas Urquhart) that the piece of artillery was actually tried upon a plain in Scotland with complete success, a number of sheep and cattle being destroyed.

In 1614 appeared the work which in the history of British science can be placed as second only to Newton’s Principia. The full title is as follows: Mirifici Logarithmorum Canonis descriptio, Ejusque usus, in utraque Trigonometria; ut etiam in omni Logistica Mathematica, Amplissimi, Facillimi, & expeditissimi explicatio. Authore ac Inventore Ioanne Nepero, Barone Merchistonii, &c., Scoto. Edinburgi, ex officinâ Andreae Hart Bibliopolae, CIↃ.DC.XIV. This is printed on an ornamental title-page. The work is a small-sized quarto, containing fifty-seven pages of explanatory matter and ninety pages of tables.

The nature of logarithms is explained by reference to the motion of points in a straight line, and the principle upon which they are based is that of the correspondence of a geometrical and an arithmetical series of numbers. The table gives the logarithms of sines for every minute to seven figures. This work contains the first announcement of logarithms to the world, the first table of logarithms and the first use of the name logarithm, which was invented by Napier.

In 1617 Napier published his Rabdologia,^[5] a duodecimo of one hundred and fifty-four pages; there is prefixed to it as preface a dedicatory epistle to the high chancellor of Scotland. The method which Napier terms “Rabdologia” consists in the use of certain numerating rods for the performance of multiplications and divisions. These rods, which were commonly called “Napier’s bones,” will be described further on. The second method, which he calls the “Promptuarium Multiplicationis” on account of its being the most expeditious of all for the performance of multiplications, involves the use of a number of lamellae or little plates of metal disposed in a box. In an appendix of forty-one pages he gives his third method, “local arithmetic,” which is performed on a chess-board, and depends, in principle, on the expression of numbers in the scale of radix 2. In the Rabdologia he gives the chronological order of his inventions. He speaks of the canon of logarithms as “a me longo tempore elaboratum.” The other three methods he devised for the sake of those who would prefer to work with natural numbers; and he mentions that the promptuary was his latest invention. In the preface to the appendix containing the local arithmetic he states that, while devoting all his leisure to the invention of these abbreviations of calculation, and to examining by what methods the toil of calculation might be removed, in addition to the logarithms, rabdologia and promptuary, he had hit upon a certain tabular arithmetic, whereby the more troublesome operations of common arithmetic are performed on an abacus or chess-board, and which may be regarded as an amusement rather than a labour, for, by means of it, addition, subtraction, multiplication, division and even the extraction of roots are accomplished simply by the motion of counters. He adds that he has appended it to the Rabdologia, in addition to the promptuary, because he did not wish to bury it in silence nor to publish so small a matter by itself. With respect to the calculating rods, he mentions in the dedication that they had already found so much favour as to be almost in common use, and even to have been carried to foreign countries; and that he has been advised to publish his little work relating to their mechanism and use, lest they should be put forth in some one else’s name.

John Napier died on the 4th of April 1617, the same year as that in which the Rabdologia was published. His will, which is extant, was signed on the fourth day before his death. No particulars are known of his last illness, but it seems likely that death came upon him rather suddenly at last. In both the Canonis descriptio and the Rabdologia, however, he makes reference to his ill-health. In the dedication of the former he refers to himself as “mihi jam morbis pene confecto,” and in the “Admonitio” at the end he speaks of his “infirma valetudo”; while in the latter he says he has been obliged to leave the calculation of the new canon of logarithms to others “ob infirmam corporis nostri valetudinem.”

It has been usually supposed that John Napier was buried in St Giles’s church, Edinburgh, which was certainly the burial-place of some of the family, but Mark Napier (Memoirs, p. 426) quotes Professor William Wallace, who, writing in 1832, gives strong reasons for believing that he was buried in the old church of St Cuthbert.

Professor Wallace’s words are—

“My authority for this belief is unquestionable. It is a Treatise on Trigonometry, by a Scotsman, James Hume of Godscroft, Berwickshire, a place still in possession of the family of Hume. The work in question, which is rare, was printed at Paris, and has the date 1636 on the title-page, but the royal privilege which secured it to the author is dated in October 1635, and it may have been written several years earlier. In his treatise (page 116) Hume says, speaking of logarithms, ‘L’inuenteur estoit un Seigneur de grande condition, et duquel la posterité est aujourd’huy en possession de grandes dignitéz dans le royaume, qui estant sur l’age, et grandement trauaillé des gouttes ne pouvait faire autre chose que de s’adonner aux sciences, et principalment aux mathematiques et à la logistique, à quoy il se plaisoit infiniment, et auec estrange peine, a construict ses Tables des Logarymes, imprimees à Edinbourg en l’an 1614. . . . Il mourut l’an 1616, et fut enterré hors la Porte Occidentale d’Edinbourg, dans l’Eglise de Sainct Cudbert.’ ”

There can be no doubt that Napier’s devotion to mathematics was not due to old age and the gout, and that he died in 1617 and not in 1616; still these sentences were written within eighteen years of Napier’s death, and their author seems to have had some special sources of information. Additional probability is given to Hume’s assertion by the fact that Merchiston is situated in St Cuthbert’s parish. It is nowhere else recorded that Napier suffered from the gout. It has been stated that Napier’s mathematical pursuits led him to dissipate his means. This is not so, for his will (Memoirs, p. 427) shows that besides his large estates he left a considerable amount of personal property.

The Canonis Descriptio on its publication in 1614, at once attracted the attention of Edward Wright, whose name is known in connexion with improvements in navigation, and Henry Briggs, then professor of geometry at Gresham College, London. The former translated the work into English, but he died in 1615, and the translation was published by his son Samuel Wright in 1616. Briggs was greatly excited by Napier’s invention and visited him at Merchiston in 1615, staying with him a whole month; he repeated his visit in 1616 and, as he states, “would have been glad to make him a third visit if it had pleased God to spare him so long.” The logarithms introduced by Napier in the Descriptio are not the same as those now in common use, nor even the same as those now called Napierian or hyperbolic logarithms. The change from the original logarithms to common or decimal logarithms was made by both Napier and Briggs, and the first tables of decimal logarithms were calculated by Briggs, who published a small table, extending to 1000, in 1617, and a large work, Arithmetica Logarithmica,^[6] containing logarithms of numbers to 30,000 and from 90,000 to 100,000, in 1624. (See Logarithm.)

Napier’s Descriptio of 1614 contains no explanation of the manner in which he had calculated his table. This account he kept back, as he himself states, in order to see from the reception met with by the Descriptio, whether it would be acceptable. Though written before the Descriptio it had not been prepared for press at the time of his death, but was published by his son Robert in 1619 under the title Mirifici Logarithmorum Canonis Constructio.^[7]In this treatise (which was written before Napier had invented the name logarithm) logarithms are called “artificial numbers.”

The different editions of the Descriptio and Constructio, as well as the reception of logarithms on the continent of Europe, and especially by Kepler, whose admiration of the invention almost equalled that of Briggs, belong to the history of logarithms (q.v.). It may, however, be mentioned here that an English translation of the Constructio of 1619 was published by W. R. Macdonald at Edinburgh in 1889, and that there is appended to this edition a complete catalogue of all Napier’s writings, and their various editions and translations, English and foreign, all the works being carefully collated, and references being added to the various public libraries in which they are to be found.

Napier’s priority in the publication of the logarithms is unquestioned and only one other contemporary mathematician seems to have conceived the idea on which they depend. There is no anticipation or hint to be found in previous writers,^[8] and it is very remarkable that a discovery or invention which was to exert so important and far-reaching an influence on astronomy and every science involving calculation was the work of a single mind.

The more one considers the condition of science at the time, and the state of the country in which the discovery took place, the more wonderful does the invention of logarithms appear. When algebra had advanced to the point where exponents were introduced, nothing would be more natural than that their utility as a means of performing multiplications and divisions should be remarked; but it is one of the surprises in the history of science that logarithms were invented as an arithmetical improvement years before their connexion with exponents was known. It is to be noticed also that the invention was not the result of any happy accident. Napier deliberately set himself to abbreviate multiplications and divisions—operations of so fundamental a character that it might well have been thought that they were in rerum natura incapable of abbreviation; and he succeeded in devising, by the help of arithmetic and geometry alone, the one great simplification of which they were susceptible—a simplification to which nothing essential has since been added.

When Napier published the Canonis Descriptio England had taken no part in the advance of science, and there is no British author of the time except Napier whose name can be placed in the same rank as those of Copernicus, Tycho Brahe, Kepler, Galileo, or Stevinus. In England, Robert Recorde had indeed published his mathematical treatises, but they were of trifling importance and without influence on the history of science. Scotland had produced nothing, and was perhaps the last country in Europe from which a great mathematical discovery would have been expected. Napier lived, too, not only in a wild country, which was in a lawless and unsettled state during most of his life, but also in a credulous and superstitious age. Like Kepler and all his contemporaries he believed in astrology, and he certainly also had some faith in the power of magic, for there is extant a deed written in his own handwriting containing a contract between himself and Robert Logan of Restalrig, a turbulent baron of desperate character, by which Napier undertakes “to serche and sik out, and be al craft and ingyne that he dow, to tempt, trye, and find out” some buried treasure supposed to be hidden in Logan’s fortress at Fastcastle, in consideration of receiving one-third part of the treasure found by his aid. Of this singular contract, which is signed, “Robert Logane of Restalrige” and “Jhone Neper, Fear of Merchiston,” and is dated July 1594, a facsimile is given in Mark Napier’s Memoirs. As the deed was not destroyed, but is in existence now, it is to be presumed that the terms of it were not fulfilled; but the fact that such a contract should have been drawn up by Napier himself affords a singular illustration of the state of society and the kind of events in the midst of which logarithms had their birth. Considering the time in which he lived, Napier is singularly free from superstition: his Plaine Discovery relates to a method of interpretation which belongs to a later age; he shows no trace of the extravagances which occur everywhere in the works of Kepler; and none of his writings contain allusions to astrology or magic.

After Napier’s death his manuscripts and notes came into the possession of his second son by his second marriage, Robert, who edited the Constructio; and Colonel Milliken Napier, Robert’s lineal male representative, was still in the possession of many of these private papers at the close of the 18th century. On one occasion when Colonel Napier was called from home on foreign service, these papers, together with a portrait of John Napier and a Bible with his autograph, were deposited for safety in a room of the house at Milliken, in Renfrewshire. During the owner’s absence the house was burned to the ground, and all the papers and relics were destroyed. The manuscripts had not been arranged or examined, so that the extent of the loss is unknown. Fortunately, however, Robert Napier had transcribed his father’s manuscript De Arte Logistica, and the copy escaped the fate of the originals in the manner explained in the following note, written in the volume containing them by Francis, seventh Lord Napier: “John Napier of Merchiston, inventor of the logarithms, left his manuscripts to his son Robert, who appears to have caused the following pages to have been written out fair from his father’s notes, for Mr Briggs, professor of geometry at Oxford. They were given to Francis, the fifth Lord Napier, by William Napier of Culcreugh, Esq., heir-male of the above-named Robert. Finding them in a neglected state, amongst my family papers, I have bound them together, in order to preserve them entire.—Napier, 7th March 1801.”

An account of the contents of these manuscripts was given by Mark Napier in the appendix to his Memoirs of John Napier, and the manuscripts themselves were edited in their entirety by him in 1839 under the title De Arte Logistica Joannis Naperi Merchistonii Baronis Libri qui supersunt. Impressum Edinburgi m.dccc.xxx.ix., as one of the publications of the Bannatyne Club. The treatise occupies one hunclied and sixty-two pages, and there is an introduction by Mark Napier of ninety-four pages. The Arithmetic consists of three books, entitled—(1) De Computationibus Quantitatum omnibus Logisticae speciebus communium; (2) De Logistica Arithmetica; (3) De Logistica Geometrica. At the end of this book: occurs the note—“I could find no more of this geometricall pairt amongst all his fragments.” The Algebra Joannis Naperi Merchistonii Baronis consists of two books: (1) “De nominata Algebrae parte; (2) De positiva sive cossica Algebrae parte,” and concludes with the words, “There is no more of his algebra orderlie sett doun.” The transcripts are entirely in the handwriting of Robert Napier himself, and the two notes that have been quoted prove that they were made from Napier’s own papers. The title, which is written on the first leaf, and is also in Robert Napier’s writing, runs thus: “The Baron of Merchiston his booke of Arithmeticke and Algebra. For Mr Henrie Briggs, Professor of Geometrie at Oxforde.”

These treatises were probably composed before Napier had invented the logarithms or any of the apparatuses described in the Rabdologia; for they contain no allusion to the principle of logarithms, even where we should expect to find such reference, and the one solitary sentence where the Rabdologia is mentioned (“sive omnium facillime per ossa Rhabdologiae nostrae”) was probably added afterwards. It is worth while to notice that this reference occurs in a chapter “De Multiplicationis et Partitionis compendiis miscellaneis,” which, supposing the treatise to have been written in Napier’s younger days, may have been his earliest production on a subject over which his subsequent labours were to exert so enormous an influence.

Napier uses abundantes and defectivae for positive and negative, defining them as meaning greater or less than nothing (“Abundantes sunt quantitates majores nihilo: defectivae sunt quantitates minores nihilo”). The same definitions occur also in the Canonis Descriptio (1614), p. 5: “Logarithmos sinuum, qui semper majores nihilo sunt, abundantes vocamus, et hoc signo +, aut nullo praenotamus. Logarithmos autem minores nihilo defectivos vocamus, praenotantes eis hoc signum −.” Napier may thus have been the first to use the expression “quantity less than nothing.” He uses “radicatum” for power (for root, power, exponent, his words are radix, radicatum, index).

Apart from the interest attaching to these manuscripts as the work of Napier, they possess an independent value as affording evidence of the exact state of his algebraical knowledge at the time when logarithms were invented. There is nothing to show whether the transcripts were sent to Briggs as intended and returned by him, or whether they were not sent to him. Among the Merchiston papers is a thin quarto volume in Robert Napier’s writing containing a digest of the principles of alchemy; it is addressed to his son, and on the first leaf there are directions that it is to remain in his charter-chest and be kept secret except from a few. This treatise and the transcripts seem to be the only manuscripts which have escaped destruction.

The principle of “Napier’s bones” may be easily explained by imagining ten rectangular slips of cardboard, each divided into nine squares. In the top squares of the slips the ten digits are written, and each slip contains in its nine squares the first nine multiples of the digit which appears in the top square. With the exception of the top squares, every square is divided into two parts by a diagonal, the units being written on one side and the tens on the other, so that when a multiple consists of two figures they are separated by the diagonal. Fig. 1 shows the slips corresponding to the numbers 2, 0, 8, 5 placed side by side in contact with one another, and next to them is placed another slip containing, in squares without diagonals, the first nine digits. The slips thus placed in contact give the multiples of the number 2085, the digits in each parallelogram being added together; for example, corresponding to the number 6 on the right-hand slip, we have 0, 8+3, 0+4, 2, 1; whence we find 0, 1, 5, 2, 1 as the digits, written backwards, of 6×2085. The use of the slips for the purpose of multiplication is now evident; thus to multiply 2085 by 736 we take out in this manner the multiples corresponding to 6, 3, 7, and set down the digits as they are obtained, from right to left, shifting them back one place and adding up the columns as in ordinary multiplication, viz. the figures as written down are—

12510

6255

14595

1534560

Napier’s rods or bones consist of ten oblong pieces of wood or other material with square ends. Each of the four faces of each rod contains multiples of one of the nine digits, and is similar to one of the slips just described, the first rod containing the multiples of 0, 1, 9, 8, the second of 0, 2, 9, 7, the third of 0, 3, 9, 6, the fourth of 0, 4, 9, 5, the fifth of 1, 2, 8, 7, the sixth of 1, 3, 8, 6, the seventh of 1, 4, 8, 5, the eighth of 2, 3, 7, 6, the ninth of 2, 4, 7, 5, and the tenth of 3, 4, 6, 5. Each rod therefore contains on two of its faces multiples of digits which are complementary to those on the other two faces; and the multiples of a digit and of its complement are reversed in position. The arrangement of the numbers on the rods will be evident from fig. 2, which represents the four faces of the fifth rod. The set of ten rods is thus equivalent to four sets of slips as described above, and by their means we may multiply every number less than 11,111, and also any number (consisting of course of not more than ten digits) which can be formed by the top digits of the bars when placed side by side. Of course two sets of rods may be used, and by their means we may multiply every number less than 111,111,111 and so on. It will be noticed that the rods only give the multiples of the number which is to be multiplied, or of the divisor when they are used for division, and it is evident that they would be of little use to any one who knew the multiplication table as far as 9×9. In multiplications or divisions of any length it is generally convenient to begin by forming a table of the first nine multiples of the multiplicand or divisor, and Napier’s bones at best merely provide such a table, and in an incomplete form, for the additions of the two figures in the same parallelogram have to be performed each time the rods are used. The Rabdologia attracted more general attention than the logarithms, and as has been mentioned, there were several editions on the Continent. Nothing shows more clearly the rude state of arithmetical knowledge at the beginning of the 17th century than the universal satisfaction with which Napier’s invention was welcomed by all classes and regarded as a real aid to calculation. Napier also describes in the Rabdologia two other larger rods to facilitate the extraction of square and cube roots. In the Rabdologia the rods are called “virgulae,” but in the passage quoted above from the manuscript on arithmetic they are referred to as “bones” (ossa).

Besides the logarithms and the calculating rods or bones, Napier’s name is attached to certain rules and formulae in spherical trigonometry. “Napier’s rules of circular parts,” which include the complete system of formulae for the solution of right-angled triangles, may be enunciated as follows. Leaving the right angle out of consideration, the sides including the right angle, the complement of the hypotenuse, and the complements of the other angles are called the circular parts of the triangle. Thus there are five circular parts, a, b, 90°—,A 90°—c, 90°—B, and these are supposed to be arranged in this order (i.e. the order in which they occur in the triangle) round a circle. Selecting any part and calling it the middle part, the two parts next it are called the adjacent parts and the remaining two parts the opposite parts. The rules then are—

sine of the middle part = product of tangents of adjacent parts
sine of the middle part = product of cosines of opposite parts.

These rules were published in the Canonis Descriptio (1614), and Napier has there given a figure, and indicated a method, by means of which they may be proved directly. The rules are curious and interesting, but of very doubtful utility, as the formulae are best remembered by the practical calculator in their unconnected form.

“Napier’s analogies” are the four formulae—

$\tan {\tfrac {1}{2}}\left({\mbox{A}}+{\mbox{B}}\right)={\frac {\cos {\tfrac {1}{2}}\left(a-b\right)}{\cos {\tfrac {1}{2}}\left(a+b\right)}}\cot {\tfrac {1}{2}}{\mbox{C}}$,	$\tan {\tfrac {1}{2}}\left({\mbox{A}}-{\mbox{B}}\right)={\frac {\sin {\tfrac {1}{2}}\left(a-b\right)}{\sin {\tfrac {1}{2}}\left(a+b\right)}}\cot {\tfrac {1}{2}}{\mbox{C}}$;
$\tan {\tfrac {1}{2}}\left(a+b\right)={\frac {\cos {\tfrac {1}{2}}\left({\mbox{A}}-{\mbox{B}}\right)}{\cos {\tfrac {1}{2}}\left({\mbox{A}}+{\mbox{B}}\right)}}\tan {\tfrac {1}{2}}c$,	$\tan {\tfrac {1}{2}}\left(a-b\right)={\frac {\cos {\tfrac {1}{2}}\left({\mbox{A}}-{\mbox{B}}\right)}{\cos {\tfrac {1}{2}}\left({\mbox{A}}+{\mbox{B}}\right)}}\tan {\tfrac {1}{2}}c$.

They were first published after his death in the Constructio among the formulae in spherical trigonometry, which were the results of his latest work. Robert Napier says that these results would have been reduced to order and demonstrated consecutively but for his father’s death. Only one of the four analogies is actually given by Napier, the other three being added by Briggs in the remarks which are appended to Napier’s results. The work left by Napier is, however, rough and unfinished, and it is uncertain whether he knew of the other formulae or not. They are, however, so simply deducible from the results he has given that all the four analogies may be properly called by his name. An analysis of the formulae contained in the Descriptio and Constructio is given by Delambre in vol. i. of his Histoire de l’Astronomie moderne.

To Napier seems to be due the first use of the decimal point in arithmetic. Decimal fractions were first introduced by Stevinus in his tract La Disme, published in 1585, but he used cumbrous exponents (numbers enclosed in circles) to distinguish the different denominations, primes, seconds, thirds, &c. Thus, for example, he would have written 123·456 as 123⓪4①5②6③. In the Rabdologia Napier gives an “Admonitio pro Decimali Arithmetica,” in which he commends the fractions of Stevinus and gives an example of their use, the division of 861094 by 432. The quotient is written 1993,273 in the work, and 1993,2′ 7″ 3‴ in the text. This single instance of the use of the decimal point in the midst of an arithmetical process, if it stood alone, would not suffice to establish a claim for its introduction, as the real introducer of the decimal point is the person who first saw that a point or line as separator was all that was required to distinguish between the integers and fractions, and used it as a permanent notation and not merely in the course of performing an arithmetical operation. The decimal point is, however, used systematically in the Constructio (1619), there being perhaps two hundred decimal points altogether in the book.

The decimal point is defined on p. 6 of the Constructio in the words: “In numeris periodo sic in se distinctis, quicquid post periodum notatur fractio est, cujus denominator est unitas cum tot cyphris post se, quot sunt figurae post periodum. Ut 10000000·04 valet idem, quod 100000004/100. Item 25·803, idem quod 25803/1000 Item 9999998·0005021, idem valet quod 99999985021/10000000, & sic de caeteris.” On p. 8, 10·502 is multiplied by 3·216, and the result found to be 33·774432; and on pp. 23 and 24 occur decimals not attached to integers, viz. ·4999712 and ·0004950. These examples show that Napier was in possession of all the conventions and attributes that enable the decimal point to complete so symmetrically our system of notation, viz. (1) he saw that a point or separatrix was quite enough to separate integers from decimals, and that no signs to indicate primes, seconds, &c., were required; (2) he used ciphers after the decimal point and preceding the first significant figure; and (3) he had no objection to a decimal standing by itself without any integer. Napier thus had complete command over decimal fractions and the use of the decimal point. Briggs also used decimals, but in a form not quite so convenient as Napier. Thus he prints 63·0957379 as 630957379, viz. he prints a bar under the decimals; this notation first appears without any explanation in his “Lucubrationes” appended to the Constructio. Briggs seems to have used the notation all his life, but in writing it, as appears from manuscripts of his, he added also a small vertical line just high enough to fix distinctly which two figures it was intended to separate: thus he might have written 63 0957379. The vertical line was printed by Oughtred and some of Briggs’s successors. It was a long time before decimal arithmetic came into general use, and all through the 17th century exponential marks were in common use. There seems but little doubt that Napier was the first to make use of a decimal separator, and it is curious that the separator which he used, the point, should be that which has been ultimately adopted, and after a long period of partial disuse.

The hereditary office of king’s poulterer (Pultrie Regis) was for many generations in the family of Merchiston, and descended to John Napier. The office, Mark Napier states, is repeatedly mentioned in the family charters as appertaining to the “pultre landis” near the village of Dene in the shire of Linlithgow. The duties were to be performed by the possessor or his deputy; and the king was entitled to demand the yearly homage of a present of poultry from the feudal holder. The pultrelands and the office were sold by John Napier in 1610 for 1700 marks. With the exception of the pultrelands all the estates he inherited descended to his posterity.

With regard to the spelling of the name, Mark Napier states that among the family papers there exist a great many documents signed by John Napier. His usual signature was “Jhone Neper,” but in a letter written in 1608, and in all deeds signed after that date, he wrote “Jhone Nepair.” His letter to the king prefixed to the Plaine Discovery is signed “John Napeir.” His own children, who sign deeds along with him, use every mode except Napier, the form now adopted by the family, and which is comparatively modern. In Latin he always wrote his name “Neperus.” The form “Neper” is the oldest, as John, third Napier of Merchiston, so spelt it in the 15th century.

Napier frequently signed his name “Jhone Neper, Fear of Merchiston.” He was “Fear of Merchiston” because, more majorum, he had been invested with the fee of his paternal barony during the lifetime of his father, who retained the liferent. He has been sometimes erroneously called “Peer of Merchiston,” and in the 1645 edition of the Plaine Discovery he is so styled (see Mark Napier’s Memoirs, pp. 9 and 173, and Libri qui supersunt, p. xciv.).

The bibliography of Napier’s work attached to W. R. Macdonald’s translation of the Canonis Constructio (1889) is complete and valuable. Napier’s three mathematical works are reprinted by N. L. W. A. Gravelaar in Verhandelingen der Kon. Akad. van Wet te Amsterdam, 1. sectie, deel 6 (1899).  (J. W. L. G.) 

---

1. The descent of the first Napier of Merchiston has been traced to “Johan le Naper del Counte de Dunbretan,” who was one of those who swore fealty to Edward I. in 1296 and defended the castle of Stirling against him in 1304; but there is no authority for this genealogy. The legend with regard to the origin of the name Napier was given by Sir Alexander Napier, eldest son of John Napier, in 1625, in these words: “One of the ancient earls of Lennox in Scotland had issue three sons: the eldest, that succeeded him to the earldom of Lennox; the second, whose name was Donald; and the third, named Gilchrist. The then king of Scotland having wars, did convocate his lieges to battle, amongst whom that was commanded was the earl of Lennox, who, keeping his eldest son at home, sent his two sons to serve for him with the forces that were under his command. . . . After the battle, as the manner is, every one drawing and setting forth his own acts, the king said unto them, ye have all done valiantly, but there is one amongst you who hath Na-Peer (i.e. no equal); and calling Donald into his presence commanded him, in regard to his worthy service, and in augmentation of his honour, to change his name from Lennox to Napier, and gave him the lands of Gosford, and lands in Fife, and made him his own servant, which discourse is confirmed by evidences of mine, wherein we are called Lennox alias Napier.”
2. A Plaine Discovery of the whole Revelation of Saint Iohn: set downe in two treatises: The one searching and proving the true interpretation thereof: The other applying the same paraphrastically and historically to the text. Set foorth by John Napier L. of Marchistoun younger. Whereunto are annexed certaine Oracles of Sibylla, agreeing with the Revelation and other places of Scripture. Edinburgh, printed by Robert Walde-grave, printer to the King’s Majestie, 1593. Cum privilegio Regali.
3. A Dutch translation was published at Middelburg in 1600 and a second edition in 1607. The work was translated into French by George Thomson, a naturalized Scotsman residing in La Rochelle, and published by him at that town in 1602, under the title Ouverture de tous les secrets de l’Apocalypse. . . . Par Jean Napeir (c. a. d.) Nonpareil, Sieur de Merchiston, reveue par lui-mesme, et mise en François par Georges Thomson, Escossois. Subsequent editions were published in 1603, 1605 and 1607. German translations were published at Gera in 1611 and at Frankfort in 1605 and 1627. The second edition in English appeared at Edinburgh in 1611, and in the preface to it Napier states he intended to have published an edition in Latin soon after the original publication in 1593, but that, as the work had now been made public by the French and Dutch translations, besides the English editions, and as he was “advertised that our papistical adversaries wer to write larglie against the said editions that are alreadie set out,” he defers the Latin edition “till having first seene the adversaries objections, I may insert in the Latin edition an apologie of that which is rightly done, and an amends of whatsoever is amisse.” No criticism on the work was published, and there was no Latin edition. A third edition appeared at Edinburgh in 1645. Corresponding to the first two Edinburgh editions, copies were issued bearing the London imprint and dates 1594 and 1611.
4. A facsimile of this document is given by Mark Napier in his Memoirs of John Napier (1834), p. 248.
5. Rabdologiae, seu Numerationis per virgulas Libri duo: Cum Appendice de expeditissimo Multiplicationis promptuario. Quibus accessit & Arithmeticae Localis Liber unus. Authore & Inventore Ioanne Nepero, Barone Merchistonii, &c., Scoto. Edinburgi, Excudebat Andreas Hart (1617). Foreign editions were published in Italian at Verona in 1623, in Latin at Leiden in 1626 and 1628, and in Dutch at Gouda in 1626. In 1623 Ursinus published Rhabdologia Neperiana at Berlin, and the rods or bones were described in several other works.
6. The title runs as follows: Arithmetica Logarithmica, sive Logarithmorum chiliades triginta. . . . Hos numeros primus invenit clarissimus vir Iohannes Neperus Baro Merchistonij; eos autem ex eiusdem sententia mutavit, eorumque ortum et usum illustravit Henricus Briggius. . . .
7. The full title was: Mirifici Logarithmorum Canonis Constructio; Et eorum ad naturales ipsorum numeros habitudines; unà cum Appendice, de aliâ eâque praestantiore Logarithmorum specie condendâ. Quibus accessere Propositiones ad triangula sphaerica faciliore calculo resolvenda: Unà cum Annotationibus aliquot doctissimi D. Henrici Briggii, in eas & memoratam appendicem. Authore & Inventore Ioanne Nepero, Barone Merchistonii, &c. Scoto. Edinburgi, Excudebat Andreas Hart, Anno Domini 1619. There is also preceding this title-page an ornamental title-page, similar to that of the Descriptio of 1614; the words are different, however, and run—Mirifici Logarithmorum Canonis Descriptio . . . Accesserunt Opera Posthuma: Primò, Mirifici ipsius canonis constructio, & Logarithmorum ad naturales ipsorum numeros habitudines. Secundò, Appendix de aliâ, eâque praestantiore Logarithmorum specie construenda. Tertiò, Propositiones quaedam eminentissimae, ad Triangula sphaerica mirâ facilitate resolvenda. . . . It would appear that this title-page was to be substituted for the title-page of the Descriptio of 1614 by those who bound the two books together.
8. The work of Justus Byrgius is described in the article Logarithm. In that article it is mentioned that a Scotsman in 1594 in a letter to Tycho Brahe held out some hope of logarithms; it is likely that the person referred to is John Craig, son of Thomas Craig, who has been mentioned as one of the colleagues of John Napier’s father as justice-depute.